A Herschel [C II] Galactic plane survey III. [C II] as a tracer of star formation

Context. The [C II] 158 mu m line is the brightest far-infrared cooling line in galaxies, representing 0.1 to 1% of their far-infrared continuum emission, and is therefore a potentially powerful tracer of star formation activity. The [C II] line traces different phases of the interstellar medium (ISM), including the diffuse ionized medium, warm and cold atomic clouds, clouds in transition from atomic to molecular, and dense and warm photon dominated regions (PDRs). Therefore without being able to separate the contributions to the [C II] emission, the relationship of this fine structure line emission to star formation has been unclear. Aims. We study the relationship between the [C II] emission and the star formation rate (SFR) in the Galactic plane and separate the relationship of different ISM phases to the SFR. We compare these relationships to those in external galaxies and local clouds, allowing examinations of these relationships over a wide range of physical scales. Methods. We compare the distribution of the [C II] emission, with its different contributing ISM phases, as a function of Galactocentric distance with the SFR derived from radio continuum observations. We also compare the SFR with the surface density distribution of atomic and molecular gas, including the CO-dark H-2 component. Results. The [C II] and SFR are well correlated at Galactic scales with a relationship that is in general agreement with that found for external galaxies. By combining [C II] and SFR data points in the Galactic plane with those in external galaxies and nearby star forming regions, we find that a single scaling relationship between the [C II] luminosity and SFR applies over six orders of magnitude. The [C II] emission from different ISM phases are each correlated with the SFR, but only the combined emission shows a slope that is consistent with extragalactic observations. These ISM components have roughly comparable contributions to the Galactic [C II] luminosity: dense PDRs (30%), cold H I (25%), CO-dark H-2 (25%), and ionized gas (20%). The SFR-gas surface density relationship shows a steeper slope compared to that observed in galaxies, but one that it is consistent with those seen in nearby clouds. The different slope is a result of the use of a constant CO-to-H-2 conversion factor in the extragalactic studies, which in turn is related to the assumption of constant metallicity in galaxies. We find a linear correlation between the SFR surface density and that of the dense molecular gas.